10,828 research outputs found
Maximising the number of induced cycles in a graph
We determine the maximum number of induced cycles that can be contained in a
graph on vertices, and show that there is a unique graph that
achieves this maximum. This answers a question of Tuza. We also determine the
maximum number of odd or even cycles that can be contained in a graph on vertices and characterise the extremal graphs. This resolves a conjecture
of Chv\'atal and Tuza from 1988.Comment: 36 page
On the inducibility of cycles
In 1975 Pippenger and Golumbic proved that any graph on vertices admits
at most induced -cycles. This bound is larger by a
multiplicative factor of than the simple lower bound obtained by a blow-up
construction. Pippenger and Golumbic conjectured that the latter lower bound is
essentially tight. In the present paper we establish a better upper bound of
. This constitutes the first progress towards proving
the aforementioned conjecture since it was posed
Maximising -Colourings of Graphs
For graphs and , an -colouring of is a map
such that . The number of -colourings of is denoted by .
We prove the following: for all graphs and , there is a
constant such that, if , the graph
maximises the number of -colourings among all
connected graphs with vertices and minimum degree . This answers a
question of Engbers.
We also disprove a conjecture of Engbers on the graph that maximises the
number of -colourings when the assumption of the connectivity of is
dropped.
Finally, let be a graph with maximum degree . We show that, if
does not contain the complete looped graph on vertices or as a
component and , then the following holds: for
sufficiently large, the graph maximises the number of
-colourings among all graphs on vertices with minimum degree .
This partially answers another question of Engbers
Proof of Koml\'os's conjecture on Hamiltonian subsets
Koml\'os conjectured in 1981 that among all graphs with minimum degree at
least , the complete graph minimises the number of Hamiltonian
subsets, where a subset of vertices is Hamiltonian if it contains a spanning
cycle. We prove this conjecture when is sufficiently large. In fact we
prove a stronger result: for large , any graph with average degree at
least contains almost twice as many Hamiltonian subsets as ,
unless is isomorphic to or a certain other graph which we
specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ
MAXIMISING THE NUMBER OF CONNECTED INDUCED SUBGRAPHS OF UNICYCLIC GRAPHS
Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g and k pendent vertices. In this paper, we give a partial characterisation of the structure of all maximal graphs in G(n, d, g, k) for the number of connected induced subgraphs. For the special case d = 1, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the maximum number of connected induced subgraphs given: (1) order, girth, and number of pendent vertices; (2) order and girth; (3) order
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